Spectral conditions on the state of a composite quantum system implying its separability
G A Raggio · Journal of Physics A Mathematical and General · 2005
The separability modulus ℓ(ρ) of a state ρ of an arbitrary finite composite quantum system is the largest t in [0, 1] such that t ⋅ ρ + (1 − t ) ⋅ τ is separable, where τ is the normalized trace. The basic properties of ℓ, introduced by Vidal and Tarrach in another guise, are briefly established. With these properties, we obtain conditions on the spectrum of a state which imply that it is separable. As a consequence, we show that for any Hamiltonian H the thermal equilibrium states e − H / T /Tr(e − H / T ) are separable if T is large enough. Also, for F a unitarily invariant, convex continuous real-valued function on states, for which F (ρ) > F (τ) whenever ρ ≠ τ, there is a critical C F such that F (ρ) ⩽ C F implies that ρ is separable, and for each possible c > C F there are entangled states ϕ with F (ϕ) = c . This class includes all strictly convex unitarily invariant continuous functions, and also every non-trivial partial eigenvalue-sum. Some C F are computed. General upper and lower bounds for C F are given, and then improved for bipartite systems.